Number System Converter

How do you convert between number bases?

To convert to decimal, multiply each digit by its place value and add the results; to convert from decimal, divide by the base repeatedly and read the remainders from last to first. Both rules work because in any positional system each place is worth a power of the base. In decimal the places are 1, 10, 100 and so on; in binary they are 1, 2, 4, 8, 16; in hexadecimal they are 1, 16, 256, 4,096.

Worked example: 156 to binary and back

Divide 156 by 2 and keep the remainders: 156 → 78 r0, 78 → 39 r0, 39 → 19 r1, 19 → 9 r1, 9 → 4 r1, 4 → 2 r0, 2 → 1 r0, 1 → 0 r1. Reading the remainders from bottom to top gives 10011100. To check, add the place values of the 1s: 128 + 16 + 8 + 4 = 156. The decimal to binary converter shows the same steps for any number, and the binary to decimal converter reverses them.

Why hex and binary convert so easily

Because 16 is 2⁴, you never need decimal as a middle step between binary and hex. Replace each hex digit with its 4-bit pattern (F = 1111, 2 = 0010), or split a binary number into groups of four from the right. This is how programmers read memory dumps and bit masks at a glance.

Why do computers use binary?

Computers use binary because their circuits are built from switches with two reliable states, on and off, which map directly to 1 and 0. A transistor that only has to tell high voltage from low voltage can tolerate noise, heat and manufacturing variation far better than one that must separate ten levels, so two states make hardware cheaper, faster and more reliable.

Binary also suits logic. The operations AND, OR and NOT from Boolean algebra work on true and false values, and every arithmetic circuit, from an adder to a full processor, is built by combining them. Numbers, text, images and sound are all stored as patterns of bits: a character in ASCII is 7 bits, a pixel in a typical screen image is 24 bits (8 each for red, green and blue), and a sample in CD audio is 16 bits.

People find long rows of ones and zeros hard to read, which is why hexadecimal exists. Four bits map to exactly one hex digit, so a 24-bit color such as 11111111 10100101 00000000 is written #FFA500, a format defined for the web by the W3C CSS Color Module. To check any bit pattern by hand, the binary to decimal converter adds up the place values for you, and the digital and typography converter covers the pixel side of screen work.

Where is each number system used?

Decimal is the system people read, binary is the one hardware runs on, and hexadecimal and octal are compact shorthands programmers use to read binary. Each shows up in a predictable set of places.

• Binary (base 2): the native language of digital hardware, used for bit flags, network subnet masks and logic design.
• Octal (base 8): Unix and Linux file permissions, where chmod 755 gives the owner read, write and execute and everyone else read and execute (binary 111 101 101).
• Decimal (base 10): everyday arithmetic, money and most user interfaces.
• Hexadecimal (base 16): memory addresses, color codes, MAC addresses, Unicode code points such as U+00E9, and byte values in debuggers.
• Roman numerals: clock faces, book chapters, film copyright years and event names. The numbers to Roman numerals converter turns 2026 into MMXXVI.

For file and memory sizes built on these bases, the data storage converter explains why 1 KiB is 1,024 bytes.

How are negative numbers stored in binary?

Computers store negative integers in two's complement, a signed format in which the highest bit carries a negative place value. To write a negative number, take the binary of its absolute value, flip every bit and add 1. In 8 bits, 5 is 00000101; flipping gives 11111010, and adding 1 gives 11111011, which is FB in hex. An 8-bit signed value ranges from −128 (10000000) to 127 (01111111), and −1 is all ones, 11111111 or FF.

This is why the same hex byte can mean two things: FF is 255 when read as unsigned and −1 when read as signed. The converter on this page treats input as an unsigned whole number, so enter the positive bit pattern when you want to see it in other bases.

How many values can a number of bits hold?

A group of n bits can represent 2ⁿ different values: 0 to 2ⁿ − 1 when unsigned, or −2ⁿ⁻¹ to 2ⁿ⁻¹ − 1 in two's complement. A bit is one binary digit, and a byte is 8 bits, so one byte holds 256 values. Each extra bit doubles the range, which is why sizes in computing climb in powers of two.

The limits have real consequences. A signed 32-bit counter of seconds since January 1, 1970, the traditional Unix timestamp, overflows at 2,147,483,647 seconds, which falls on January 19, 2038 at 03:14:07 UTC. Systems that moved to 64-bit time avoid the problem for billions of years. Converting the maximum values is also a good check on your work: 65,535 in the decimal to binary converter gives sixteen ones, and the same number in the decimal to hex converter gives FFFF.

Storage sizes follow the same powers of two. The binary prefixes kibi, mebi and gibi, standardized in IEC 80000-13, mean 2¹⁰, 2²⁰ and 2³⁰ bytes, and the data storage converter sets them against the decimal kilobyte, megabyte and gigabyte.